解:将方程分组为[(9x2-1)dx-4ydl+(ydx+xdy)=0,凑微分 得d(3x3-x-2y2)+d(xy)=0,积分得通解:3x3-x-2y2+xy=c. 5)ly sin(/y)-yr-cos(y/ x)+1]dr +[e-l cos(y/)-ry-2sin(z/y)+y-2dy=0 解法一:记M(x,y)=y-lsin(x/y)-yx-2cos(y/x)+1 N(a, y)=a cos(y/a)-ry"sin(a/y)+y 可得aM(x,y)/y=aN(x,y)/0x,因此方程是恰当的.设其积分 为U(x,y)=c,则OU(x,y/y=N(x,y),关于y积分,得 U(x,y)=/-cos(y/)-xy-2sin(x/y)+y-21y sin(y/a)-cos (a/y)-1/y+c(a) 其中c(x)是待定的x的函数为求c(x),利用恒等式 OU(x,y)/0x=M(x,y),可得c(x)=1,故可取c(x)=x.所以积分为 U(r, y)=c, Ip U(a, y)=sin(y/x)-cos(c/3)-1/y+r 解法二个将微分方程组合为sin(x/y)/ydx-rsin(x/y)/y2dyl ycos(y/x)/x2dx+cos(y/x)/rdl+dx+1/y2d=0,凑微分,积 分得个-cos(x/y)+sin(y/a)+x-1/y=c 6)2.c(y exp(ax)-1)d r + exp(a)dy 解:将原方程化为[d(exp(x2)+exp(x2)dyl-2rdr=0,凑微分得 d(y exp(x2)-d(x2)=0,积分得通解yexp(x2)-x2=c,或解出显函数 形式:y=(c+x2)exp(-x2) 7)(exp(ar)+3y )dr +2 ry dy=0 解:将方程分组成(3y2dx+2rydy)+exp(x)dx=0,凑微分得 x-2d(x3y2)+exp(x)dx=0,可见积分因子可取为x2,从而化成全微分 方程d(x3y2)+x2exp(x)dr=0,积分得通解x3y2+exp(x)(x2-2x+2)=c 8(3+y +r)dr +ry dy=0 解:将方程分组成(x2+x)dx+(y2dx+xydy)=0,凑微分得 (x2+x)dx+(2x)-1d(x23y2)=0,可见积分因子可取为12x,从而化成全 微分方程12x(x2+x)dx+6d(x23y2)=0,积分得通解3x4+4x3+6x2y2=c. 9)(a+2y) dr +ady=0 解:将方程分组成rdx+(2ydx+xdy)=0,凑微分得 rdx+x-ld(x2y)=0,可见积分因子可取为3x,从而化成全微分方程 3x2dx+3d(x2y)=0,积分得通解x3+3x23y=c 解:将方程分组成2ry2dx+(ydx-rdy)=0,凑微分得 xy2dx+y2d(x/y)=0,可见积分因子可取为y-2,从而化成全微分 方程2rdx+d(x/y)=0.积分得通解x2+x/y=c,还有特解y=0不 包含在通解中.它是原方程两边除以零而丢失的解. 11)[y-x(x2+y2)]dx-rdy=0 解:将方程分组成-x(x2+y2)dx-(xdy-ydx)=0,凑微分得 r(x2+y2)dx-(x2+y2) d arctan(y/x)=0,可见积分因子可取为 2/(x2+y2),从而化成全微分方程2xdx+2 d arctan(y/x)=0,积分得 通解x2+2 arctan(y/x): ~
1 [(9x 2 − 1) dx − 4y dy] + (y dx + x dy) = 0, G￾
v d(3x 3 − x − 2y 2 ) + d(xy) = 0, )
v= : 3x 3 − x − 2y 2 + xy = c. 5) [y −1 sin(x/y) − yx−2 cos(y/x) + 1] dx +[x −1 cos(y/x) − xy−2 sin(x/y) + y −2 ] dy = 0 : ^ M(x, y) = y −1 sin(x/y) − yx−2 cos(y/x) + 1, N(x, y) = x −1 cos(y/x) − xy−2 sin(x/y) + y −2 , Rv ∂M(x, y)/∂y = ∂N(x, y)/∂x, [_/`= . q)
1 U(x, y) = c, a ∂U(x, y)/∂y = N(x, y), yz y )
, v U(x, y) = Z [x −1 cos(y/x) − xy−2 sin(x/y) + y −2 ]∂y = sin(y/x) − cos(x/y) − 1/y + c(x) qX c(x) /b! x J. 1] c(x), X9G/ ∂U(x, y)/∂x = M(x, y), Rv c 0 (x) = 1, VRk c(x) = x. dS)
1 U(x, y) = c, qX U(x, y) = sin(y/x) − cos(x/y) − 1/y + x. N ~￾
E1 [sin(x/y)/y dx − x sin(x/y)/y2 dy] +[−y cos(y/x)/x2 dx + cos(y/x)/x dy] + [dx + 1/y2 dy] = 0, G￾
,)
vN − cos(x/y) + sin(y/x) + x − 1/y = c 6) 2x(y exp(x 2 ) − 1) dx + exp(x 2 ) dy = 0 : ~;+1 [y d(exp(x 2 )) + exp(x 2 ) dy] − 2x dx = 0, G￾
v d(y exp(x 2 )) − d(x 2 ) = 0, )
v= y exp(x 2 ) − x 2 = c, O )cJ O/: y = (c + x 2 ) exp(−x 2 ). 7) (exp(x) + 3y 2 ) dx + 2xy dy = 0 : ~
H (3y 2 dx + 2xy dy) + exp(x) dx = 0, G￾
v x −2 d(x 3y 2 )+exp(x) dx = 0, RF)
[dRk1 x 2 , uW+H]￾
d(x 3y 2 )+x 2 exp(x) dx = 0, )
v= x 3y 2+exp(x)(x 2−2x+2) = c. 8) (x 2 + y 2 + x)dx + xy dy = 0 : ~
H (x 2 + x) dx + (y 2 dx + xy dy) = 0, G￾
v (x 2+x) dx+(2x) −1 d(x 2y 2 ) = 0, RF)
[dRk1 12x, uW+H] ￾
12x(x 2+x) dx+6d(x 2y 2 ) = 0, )
v= 3x 4+4x 3+6x 2y 2 = c. 9) (x + 2y) dx + x dy = 0 : ~
H x dx + (2y dx + x dy) = 0, G￾
v x dx + x −1d(x 2y) = 0, RF)
[dRk1 3x, uW+H]￾
3x 2 dx + 3d(x 2y) = 0, )
v= x 3 + 3x 2y = c. 10) (2xy2 + y) dx − x dy = 0 : ~
H 2xy2 dx + (y dx − x dy) = 0, G￾
v 2xy2 dx + y 2 d(x/y) = 0, RF)
[dRk1 y −2 , uW+H]￾
2x dx + d(x/y) = 0, )
v= x 2 + x/y = c, >8? y = 0 Y cTA= X. 6/;wxIS$Wef . 11) [y − x(x 2 + y 2 )] dx − x dy = 0 : ~
H −x(x 2 + y 2 ) dx − (x dy − y dx) = 0, G￾
v −x(x 2 + y 2 ) dx − (x 2 + y 2 ) d arctan(y/x) = 0, RF)
[dRk1 −2/(x 2 + y 2 ), uW+H]￾
2x dx + 2d arctan(y/x) = 0, )
v = x 2 + 2 arctan(y/x) = c. 7